Number Theory Axioms Quiz

Questions and Answers

An assumed statement (correct)

A definition

A proven statement

A theorem

The whole numbers are closed under addition (correct)

The square root of a negative number is imaginary

The sum of two even numbers is even

The product of two prime numbers is prime

To prove theorems

To define terms

To solve equations

To provide a foundation for theorems (correct)

Closure axiom Signup and view all the answers

a + b = b + a Signup and view all the answers

Commutative axiom Signup and view all the answers

(a + b) + c = a + (b + c) Signup and view all the answers

Associative axiom Signup and view all the answers

a(b + c) = ab + ac Signup and view all the answers

A triangle has three sides Signup and view all the answers

Study Notes

An axiom is an assumed statement that serves as a foundation in number theory.
Axioms are not proven; they are accepted as true to establish further statements and theorems.

The closure axiom specifies that the sum of two whole numbers remains a whole number, affirming closure under addition.
The statement "the whole numbers are closed under addition" serves as an example of an axiom.

Axioms provide a foundation for theorems, supporting the structure of mathematical reasoning.
They play a critical role in defining terms and principles used throughout number theory.

The commutative axiom of addition states that changing the order of addends does not affect the sum: ( a + b = b + a ).
The associative axiom of addition allows regrouping of numbers during addition, indicated by ( (a + b) + c = a + (b + c) ).

The distributive axiom of multiplication over addition defines how multiplication interacts with addition, expressed as ( a(b + c) = ab + ac ).
This axiom enables the expansion of products of sums into sums of products.

Closure, commutativity, associativity, and distributivity are fundamental properties that guide operations in number theory.
Each axiom reflects essential relationships and behaviors in mathematics, crucial for proofs and problem-solving.